Abstract. We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group Sp(n), instead of the Lie group U(n). Denote by h a Cartan algebra of Sp(n), and ∆ the union of the zero sets of the roots of Sp(n) tensored with R 3 , each being a map from h ⊗ R 3 → R 3 . We wish to construct a map (h ⊗ R 3 )\∆ → Sp(n)/T n which is equivariant under the action of the Weyl group W n of Sp(n) (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of Sp(n), and T n is the diagonal n-torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for n = 2.

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First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every x in L = Lie(G) can be written as x = [a, b] for some a, b in L. By using the same method, we give a new proof of the following theorem (thus avoiding the classification tables of fundamental weights): in compact semisimple Lie algebras, orthogonal Cartan subalgebras always exist (where one of them can be chosen arbitrarily). Some of the consequences of this theorem are the following. (i) If L = Lie(G) is such a Lie algebra and C is any Cartan subalgebra of L, then the G-orbit of C ⊥ is all of L. (ii) The consequence in part (i) answers a question by L. Florit and W. Ziller on fatness of certain principal bundles. It also shows that in our case, the commutator map L × L → L is open at (0, 0). (iii) given any regular element x of L, there exists a regular element y such that L = [x, L] + [y, L] and x, y are orthogonal. Then we generalize this result about compact semisimple Lie algebras to the class of non-Hermitian real semisimple Lie algebras having full rank.Finally, we survey some recent related results , and construct explicitly orthogonal Cartan subalgebras in su(n), sp(n), so(n).

In this note, given a pair (g, λ), where g is a complex semisimple Lie algebra and λ ∈ h * is a dominant integral weight of g, where h ⊂ g is the real span of the coroots inside a fixed Cartan subalgebra, we associate an SU (2) and Weyl equivariant smooth map f : X → (P m (C)) n , where X ⊂ h ⊗ R 3 is the configuration space of regular triples in h, and m, n depend on the initial data (g, λ).We conjecture that, for any x ∈ X, the rank of f (x) is at least the rank of a collinear configuration in X (collinear when viewed as an ordered r-tuple of points in R 3 , with r being the rank of g). A stronger conjecture is also made using the singular values of a matrix representing f (x).This work is a generalization of the Atiyah-Sutcliffe problem to a Lie-theoretic setting.

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